Question

Is this a linear equation `5x+6y=3x-2`?

Answer 1

Yes, it is a linear equation in two variables `x` & `y`.

Explanation:

The generalized linear equation in two variables `x` & `y` is given as

`ax+by+c=0`

The above equation shows a straight line on `XY` plane.

The given equation

`5x+6y=3x-2`

or `2x+6y+2=0`

or `x+3y+1=0`

The above equation is in form of linear equation: `ax+by+c=0`

Answer 2

`" "`
Please read the explanation.

Explanation:

`" "`
Linear Equations are referred to as the equations of degree one,

where the variable's exponent ( or Power) is the value one.

Slope-Intercept Form:

This is the most common form of a linear equation.

It takes the form:

`color(red)(y=mx+b`, where

`color(red)(m` is the Slope or the Gradient

`color(red)(b` is the y-intercept

General Form:

IF there are two variables in a linear equation it takes the form:

`color(red)(Ax+By=C`

To find the x-intercept, set `color(blue)(y=0`

To find the y-intercept, set `color(blue)(x=0`

`color(blue)("Note :"`

For both of the above forms, we can construct a data table and create a graph.

Now, we will get back to the problem given to us:

`color(red)(5x+6y = 3x-2`

This is a linear equation with two variables

We can combine like terms and simplify the equation as
follows:

`rArr 5x+6y=3x-2`

Subtract `color(red)(3x` from both sides of the equation to balance the equation.

`rArr 5x+6y-3x=3x-2-3x`

`rArr 5x+6y-3x=cancel(3x)-2-cancel(3x)`

`color(blue)(rArr 2x+6y=(-2)`

Hence, we can conclude that the equation

`color(red)(5x+6y = 3x-2`

is indeed a linear equation.

y-intercept can be found by setting `color(red)(x=0`

`rArr 2x+6y=(-2)`

`rArr 2(0)+6y=(-2)`

`rArr 6y= -2`

Divide both sides by `color(red)(6`

`rArr (6y)/6= (-2)/6`

`rArr y = -1/3`

Hence we understand that `color(blue)((0,-1/3)` is the y-intercept

To find the x-intercept, set `color(red)(y=0`

`rArr 2x+6y=(-2)`

`rArr 2x+6(0)=(-2)`

`rArr 2x=-2`

Divide both sides of the equation by `color(red)(2`

`rArr (2x)/2=(-2)/2`

`rArr (cancel 2x)/cancel 2=(-2)/2`

`rArr x = -1`

Hence we understand that `color(blue)((-1,0)` is the x-intercept

We can verify these results using a graph as given below:

I hope this explanation is helpful.